The Time Traveler was confronted by a sphinx with a
faint shadow of a smile on its face.
These notes are my side of a discussion I had with James Knight on
Aumann’s Agreement theorem. I believe he is incorporating his views on the
subject in one of his books (something to look forward to!). I don’t have time
to write books myself but prefer to zip around from one subject to the next
logging my thoughts as fast as I can in order to cover as much ground as possible
in what time is left to me; there are just too many riddles to touch on! But
you never know, anyone of them might throw further inspiring light on the
meaning of life, the universe and everything!
In these notes I talk about evolution.
I have a rather qualified and reserved view on evolution as it is currently
understood. This current understanding implies that a mathematical object I refer to as the spongeam ( my name; see links below for more details) must exist in
configuration space. I personally have doubts about that. But evolution in the
weaker sense of a natural history of
change, whatever the mechanism(s) that have driven this change, has occurred over long ages. Observations
based on the messages sent to us via fossil laden rock sediment is strong
evidence of this history of change. So, when I refer to evolution below I am in
most cases thinking “natural history” rather than “spongeam”.
On the Spongeam
https://quantumnonlinearity.blogspot.com/2019/09/evolution-naked-chance.html
https://quantumnonlinearity.blogspot.com/2016/03/the-mathematics-of-spongeam.html
https://quantumnonlinearity.blogspot.com/2016/02/on-structuralism-and-spongeam.html
https://quantumnonlinearity.blogspot.com/2015/11/intelligent-designs-2001-space-odyssey.html
http://quantumnonlinearity.blogspot.com/2015/06/algorithms-searches-dualism-and_13.html
Introduction
Most Western people, as a matter of common sense, assume there’s
an invariant ontology out there that
is the depository of a common truth to be targeted and if possible converged
upon. For most people this is an axiomatic trivialism. A minority of
relativists, postmodernists, nihilists and cynics (“What is Truth?”) might
question this axiom but most common sense people take the existence of a universal
target ontology as given. Also, by and large Christians have no problem with an
ontology of “truth” because they believe God is a rational creator (although
rationality seems break down in the paranormal). But how to get to that truth
is a much more vexed question; this is the epistemological question.
What Aumann’s theorem doesn’t do is set out to prove that there is
such a thing as a Truth to agree on. That’s not what the proof shows; that’s axiomatic as far as Aumman’s
theorem is concerned, a theorem which
goes on to set a lower limit
on the computational time needed for agreement assuming there is something to
agree on. I see Aumann’s theorem as a department of epistemology because
epistemology is about developing a concept of the world on the basis of the
messages received from one’s context and that context includes messages from
other people as well as the objects around us. Aumann simplifies this process
using a relatively elementary model.
Actually it is possible to imagine circumstances where a
successful epistemology is simply not possible; e,g, where sensory data is muddled,
chaotic or only erratically available and
therefore there are then no regular
patterns to agree on (cf the paranormal). However, in these notes I’ve followed the axiomatic
assumption that there is some solid common “truth” out there to be discovered
and agreed upon.
If one accepts the common
truth axiom then in one sense Aumann’s theorem isn’t so startling or
surprising: If dispassionate observers share the same logic, share unobstructed
communications and (eventually) the same “facts” then convergence toward agreement
is an intuitively obvious conclusion; it all boils down to dispassionate observers
ultimately sharing the same set of “facts” (for presumably the logic needed to
massage those facts into a theoretical narrative is universal).
But of course human beings are far more advanced and complex
affairs than simple dispassionate machines trying to share a limited set of
facts in order to arrive at the same conclusions, as per Aumann’s theorem.
Moreover, the world human beings find themselves in doesn’t present simple
collections of facts on a plate that can then be shared willy-nilly. Fact
gathering itself can be a hard graft; the cosmos doesn’t give up its facts that
easily let alone hand them to us embedded in readymade theories.
Further; human beings are complex adaptive systems evolved to have a very complex suite of motivations
and cognitive processes that fit them for
a community life. That community life mediates both facts and theories;
this immediately puts an entirely different complexion on the matter, way in
advance of Aumann’s simple computer simulation, a simulation not programmed with the trade-offs
forced on us by social interactions and community belonging.
The world of Aumann’s theorem is a kind of toy-town cosmos not unlike the Ptolemaic cosmos – it’s a start, but its only start. I have in the past likened Aumann’s theorem to Olber’s paradox, a conclusion about starlight which proved to be wrong, but because it was wrong it pointed to the need to question the fundamental cosmological assumptions from which the paradox was derived. Olber’s paradox was therefore a profound result. Aumann’s theorem is also a profound result if it is seen in the right light.
The full copy of these notes can be found here